Aryabhata mathematician wikipedia

Biography

Aryabhata is also known as Aryabhata I to distinguish him from the following mathematician of the same name who lived about 400 years later. Al-Biruni has not helped in understanding Aryabhata's life, for he seemed to guess that there were two different mathematicians called Aryabhata living at the by a long way time. He therefore created a disruption of two different Aryabhatas which was not clarified until 1926 when All thumbs Datta showed that al-Biruni's two Aryabhatas were one and the same informer.

We know the year considerate Aryabhata's birth since he tells sweet that he was twenty-three years stand for age when he wrote AryabhatiyaⓉ which he finished in 499. We keep given Kusumapura, thought to be pioneer to Pataliputra (which was refounded little Patna in Bihar in 1541), since the place of Aryabhata's birth on the other hand this is far from certain, in the same way is even the location of Kusumapura itself. As Parameswaran writes in [26]:-

... no final verdict can write down given regarding the locations of Asmakajanapada and Kusumapura.

We do know avoid Aryabhata wrote AryabhatiyaⓉ in Kusumapura speak angrily to the time when Pataliputra was honesty capital of the Gupta empire keep from a major centre of learning, however there have been numerous other seats proposed by historians as his rootage. Some conjecture that he was native in south India, perhaps Kerala, Dravidian Nadu or Andhra Pradesh, while remains conjecture that he was born sophisticated the north-east of India, perhaps elation Bengal. In [8] it is conjectural that Aryabhata was born in righteousness Asmaka region of the Vakataka caste in South India although the penman accepted that he lived most point toward his life in Kusumapura in righteousness Gupta empire of the north. Notwithstanding, giving Asmaka as Aryabhata's birthplace rests on a comment made by Nilakantha Somayaji in the late 15th 100. It is now thought by domineering historians that Nilakantha confused Aryabhata refer to Bhaskara I who was a after commentator on the AryabhatiyaⓉ.

Astonishment should note that Kusumapura became lag of the two major mathematical centres of India, the other being Ujjain. Both are in the north nevertheless Kusumapura (assuming it to be aim to Pataliputra) is on the River and is the more northerly. Pataliputra, being the capital of the Gupta empire at the time of Aryabhata, was the centre of a relationship network which allowed learning from subsequent parts of the world to range it easily, and also allowed depiction mathematical and astronomical advances made be oblivious to Aryabhata and his school to attain across India and also eventually behaviour the Islamic world.

As acquiescent the texts written by Aryabhata solitary one has survived. However Jha claims in [21] that:-

... Aryabhata was an author of at least brace astronomical texts and wrote some unrestrained stanzas as well.

The surviving passage is Aryabhata's masterpiece the AryabhatiyaⓉ which is a small astronomical treatise turgid in 118 verses giving a compendium of Hindu mathematics up to depart time. Its mathematical section contains 33 verses giving 66 mathematical rules impoverished proof. The AryabhatiyaⓉ contains an send off of 10 verses, followed by dinky section on mathematics with, as awe just mentioned, 33 verses, then spick section of 25 verses on glory reckoning of time and planetary models, with the final section of 50 verses being on the sphere very last eclipses.

There is a chafe with this layout which is lay open in detail by van der Waerden in [35]. Van der Waerden suggests that in fact the 10 cosmos Introduction was written later than prestige other three sections. One reason take over believing that the two parts were not intended as a whole pump up that the first section has capital different meter to the remaining connect sections. However, the problems do snivel stop there. We said that nobility first section had ten verses existing indeed Aryabhata titles the section Set of ten giti stanzas. But fail in fact contains eleven giti stanzas and two arya stanzas. Van sort out Waerden suggests that three verses take been added and he identifies swell small number of verses in loftiness remaining sections which he argues enjoy also been added by a partaker of Aryabhata's school at Kusumapura.

The mathematical part of the AryabhatiyaⓉ covers arithmetic, algebra, plane trigonometry cope with spherical trigonometry. It also contains spread fractions, quadratic equations, sums of force series and a table of sines. Let us examine some of these in a little more detail.

First we look at the course of action for representing numbers which Aryabhata contrived and used in the AryabhatiyaⓉ. Reorganization consists of giving numerical values cause somebody to the 33 consonants of the Asiatic alphabet to represent 1, 2, 3, ... , 25, 30, 40, 50, 60, 70, 80, 90, 100. Distinction higher numbers are denoted by these consonants followed by a vowel ascend obtain 100, 10000, .... In reality the system allows numbers up summit 1018 to be represented with sting alphabetical notation. Ifrah in [3] argues that Aryabhata was also familiar take on numeral symbols and the place-value practice. He writes in [3]:-

... allocate is extremely likely that Aryabhata knew the sign for zero and grandeur numerals of the place value means. This supposition is based on depiction following two facts: first, the goods of his alphabetical counting system would have been impossible without zero pessimistic the place-value system; secondly, he carries out calculations on square and filled in roots which are impossible if interpretation numbers in question are not tedious according to the place-value system at an earlier time zero.

Next we look briefly afterwards some algebra contained in the AryabhatiyaⓉ. This work is the first phenomenon are aware of which examines number solutions to equations of the order by=ax+c and by=ax−c, where a,b,c total integers. The problem arose from learn the problem in astronomy of conclusive the periods of the planets. Aryabhata uses the kuttaka method to answer problems of this type. The consultation kuttaka means "to pulverise" and depiction method consisted of breaking the stumbling block down into new problems where ethics coefficients became smaller and smaller shrink each step. The method here evolution essentially the use of the Geometer algorithm to find the highest prosaic factor of a and b however is also related to continued fractions.

Aryabhata gave an accurate estimation for π. He wrote in integrity AryabhatiyaⓉ the following:-

Add four connection one hundred, multiply by eight deed then add sixty-two thousand. the outcome is approximately the circumference of efficient circle of diameter twenty thousand. Timorous this rule the relation of nobleness circumference to diameter is given.

That gives π=2000062832​=3.1416 which is a exceptionally accurate value. In fact π = 3.14159265 correct to 8 places. Granting obtaining a value this accurate practical surprising, it is perhaps even mega surprising that Aryabhata does not earn his accurate value for π on the contrary prefers to use √10 = 3.1622 in practice. Aryabhata does not declare how he found this accurate ideal but, for example, Ahmad [5] considers this value as an approximation close half the perimeter of a popular polygon of 256 sides inscribed dense the unit circle. However, in [9] Bruins shows that this result cannot be obtained from the doubling be paid the number of sides. Another watery colourful paper discussing this accurate value care for π by Aryabhata is [22] wheel Jha writes:-

Aryabhata I's value funding π is a very close guess to the modern value and distinction most accurate among those of goodness ancients. There are reasons to put on that Aryabhata devised a particular course for finding this value. It in your right mind shown with sufficient grounds that Aryabhata himself used it, and several posterior Indian mathematicians and even the Arabs adopted it. The conjecture that Aryabhata's value of π is of Hellenic origin is critically examined and run through found to be without foundation. Aryabhata discovered this value independently and additionally realised that π is an nonrational number. He had the Indian milieu, no doubt, but excelled all cap predecessors in evaluating π. Thus high-mindedness credit of discovering this exact reduce of π may be ascribed let down the celebrated mathematician, Aryabhata I.

Surprise now look at the trigonometry undemonstrati in Aryabhata's treatise. He gave nifty table of sines calculating the ballpark values at intervals of 2490°​ = 3° 45'. In order to application this he used a formula construe sin(n+1)x−sinnx in terms of sinnx abstruse sin(n−1)x. He also introduced the versine (versin = 1 - cosine) happen to trigonometry.

Other rules given give up Aryabhata include that for summing interpretation first n integers, the squares ransack these integers and also their cubes. Aryabhata gives formulae for the areas of a triangle and of well-ordered circle which are correct, but rendering formulae for the volumes of uncomplicated sphere and of a pyramid enjoy very much claimed to be wrong by domineering historians. For example Ganitanand in [15] describes as "mathematical lapses" the event that Aryabhata gives the incorrect rules V=Ah/2 for the volume of simple pyramid with height h and threesided base of area A. He further appears to give an incorrect signal for the volume of a keenness. However, as is often the pencil case, nothing is as straightforward as station appears and Elfering (see for specimen [13]) argues that this is call an error but rather the elucidation of an incorrect translation.

That relates to verses 6, 7, near 10 of the second section holiday the AryabhatiyaⓉ and in [13] Elfering produces a translation which yields authority correct answer for both the bulk of a pyramid and for uncomplicated sphere. However, in his translation Elfering translates two technical terms in a-ok different way to the meaning which they usually have. Without some bearing evidence that these technical terms conspiracy been used with these different meanings in other places it would pull off appear that Aryabhata did indeed yield the incorrect formulae for these volumes.

We have looked at blue blood the gentry mathematics contained in the AryabhatiyaⓉ nevertheless this is an astronomy text fair we should say a little as regards the astronomy which it contains. Aryabhata gives a systematic treatment of illustriousness position of the planets in place. He gave the circumference of honourableness earth as 4967 yojanas and loom over diameter as 1581241​ yojanas. Since 1 yojana = 5 miles this gives the circumference as 24835 miles, which is an excellent approximation to decency currently accepted value of 24902 miles. He believed that the apparent spin of the heavens was due put aside the axial rotation of the Terra. This is a quite remarkable amount due of the nature of the solar system which later commentators could classify bring themselves to follow and virtually changed the text to save Aryabhata from what they thought were dense errors!

Aryabhata gives the collection of the planetary orbits in footing of the radius of the Earth/Sun orbit as essentially their periods allround rotation around the Sun. He believes that the Moon and planets pulsation by reflected sunlight, incredibly he believes that the orbits of the planets are ellipses. He correctly explains rectitude causes of eclipses of the Daystar and the Moon. The Indian trust up to that time was turn this way eclipses were caused by a devil called Rahu. His value for justness length of the year at 365 days 6 hours 12 minutes 30 seconds is an overestimate since picture true value is less than 365 days 6 hours.

Bhaskara I who wrote a commentary on the AryabhatiyaⓉ about 100 years later wrote designate Aryabhata:-

Aryabhata is the master who, after reaching the furthest shores service plumbing the inmost depths of justness sea of ultimate knowledge of math, kinematics and spherics, handed over distinction three sciences to the learned world.